For smoothness objective, many existing popular message passing scheme can be derived based on gradient descent with appropriate step size choice [1]}, [2]}. However, directly computing the gradient of the fairness term makes the closed-form gradient complicated since the gradient of \(l_1\)  norm involves the sign of elements in the vector. To solve this optimization problem in a more effective and efficient manner, Fenchel conjugate [3]} is introduced to transform the original problem as bi-level optimization problem (Please see more details in Appendix ). For the general convex function \(h(\cdot )\) , its conjugate function is defined as
\(h^{*}(\mathbf {U})\stackrel{\triangle }{=}\sup \limits _{\mathbf {X}}\langle \mathbf {U}, \mathbf {X}\rangle -h(\mathbf {X})\) .
Based on Fenchel conjugate, the fairness objective can be transformed as variational representation \(h_f(\mathbf {p})=\sup \limits _{\mathbf {u}}\langle \mathbf {p},\mathbf {u} \rangle - h_f^{*}(\mathbf {u})\) , where \(\mathbf {p}=\mathbf {\Delta }_s SF(\mathbf {F})\in \mathbb {R}^{1\times d_{out}}\)  is a predicted probability vector for classification. Furthermore, the original optimization problem is equivalent to
\(\min \limits _{\mathbf {F}}\max \limits _{\mathbf {u}} h_s(\mathbf {F}) + \langle \mathbf {p},\mathbf {u} \rangle - h_f^{*}(\mathbf {u})\) 
